$L^p$-$L^q$ boundedness of pseudo-differential operators on smooth manifolds and its applications to nonlinear equations

Abstract

In this paper we study the boundedness of global pseudo-differential operators on smooth manifolds. By using the notion of global symbol we extend a classical condition of H\"ormander type to guarantee the $L^p$-$L^q$-boundedness of global operators. First we investigate $L^p$-boundedness of pseudo-differential operators in view of the H\"ormander-Mihlin condition. We also prove $L^\infty$-$BMO$ estimates for pseudo-differential operators. Later, we concentrate our investigation to settle $L^p$-$L^q$ boundedness of the Fourier multipliers and pseudo-differential operators for the range $1<p \leq 2 \leq q<\infty.$ On the way to achieve our goal of $L^p$-$L^q$ boundedness we prove two classical inequalities, namely, Paley inequality and Hausdorff-Young-Paley inequality for smooth manifolds. Finally, we present the applications of our boundedness theorems to the well-posedness properties of different types of the nonlinear partial differential equations.